Tyrion is arguably the central character of Game of Thrones, and he's certainly a fan favorite. Over the course of the eight seasons, he also went through an incredible amount of change - and was one of the few characters that even fully changed his house allegiance - a huge deal in Westeros.
He went from being the debauched younger son of a wealthy house to a hated fugitive, to the Hand of the Queen coming back to conquer Westeros, to a conflicted and miserable man, thrust into a position that he didn't want. His story takes him from the highest heights of Westeros society down to the depths and back up again - so it's no wonder that Tyrion in the final season has changed so much from the man we first met.
Drinks Far Less
Possibly Tyrion's most-quoted line is 'I drink, and I know things' - and for good reason. His alcohol consumption was staggering, even in a world where everyone seems to be swilling wine like it is going out of style. In the early seasons, Tyrion always had a drink in hand, but after his banishment (and killing his father), he actively started trying to drink himself to death. However, his relationship with Daenerys changed him, and while he could still be seen sipping a cup of wine at the end, his constant drunkenness was gone - and both literally and figuratively, Tyrion was a much more sober man.
Hair Color
The color of Tyrion's hair actually changed so much over the course of the show that fans started to comment on it! In the early seasons, Tyrion's hair was a golden blonde (very classic Lannister), and in the books it's even described as almost white. However, as the show progressed, his hair got darker and darker, until it was nearly brown. Theories abound as to why this is, from simple practicality and lighting to the idea that it represents his emotional journey into a more serious character, or even his rejection of (and by) his own family. Whatever the reason, the difference is striking.
Commitment To His Own House
Tyrion may never have been the most beloved member of the Lannisters, but at the start of the series, he was still a Lannister through and through. He had his loyalties to his family, even when he didn't particularly like them (and when they definitely didn't particularly like him), and he wore their colors, crests, and bandied the name about constantly.
By the end, though, he had completely turned against them. He murdered his own father, his sister put a price on his head, and he had become the trusted advisor of their sworn enemy.
Relationship With Jamie
Of all the Lannisters, the only one that really seemed to care about Tyrion was Jaime - but this was definitely something that fans got to see develop as the show went on. In the first season, Jaime may not have actively hated his brother the way that the rest of the family did, but as the series continued, it became clear that the two actually had a loving and meaningful relationship - from Jaime's defense of Tyrion after Joffrey's death to his help with Tyrion's escape and their final conversations, it was wonderful to watch this relationship really grow on-screen.
Facial Disfigurement
Some book fans were disappointed that this particular change was significantly less drastic than it was in the books, but it was still a huge change. After the Battle of the Blackwater, Tyrion was left with a massive scar running the length of his face (in the books he also essentially loses his nose - something that would have been practically far more difficult to create on-screen).
Willingness To Love
As we know from his history - Tyrion was originally desperate to love, although by the time that the show started, he had resigned himself to just having lots and lots of sex instead. However, when Shae came into the picture, it quickly became clear that Tyrion still desperately wanted to fall in love with her.
Her betrayal, and her decision to turn against him (and sleep with his father) had a huge impact on him. While he may have loved Daenerys, by the end of the show he was away from the brothels and from even the attempt at being with someone again.
Loses Some Of His Dragon Love
Since he was a child, Tyrion had always had a fascination with dragons - and when he was a boy, he even wanted one as a pet (just a small one, of course). However, this love of all things dragon may have faded over the course of the show, when he was facing some actual dragons thanks to his Dragon Queen. Having watched them burn down towns and eat his friend, we feel that his love and fascination isn't quite what it used to be.
Betrayed His Friends
Speaking of a dragon eating his friends, one of the biggest changes in the show was in Tyrion's relationship with Varys. These two outsiders were the closest of allies for a long time, and Varys was a huge part of Tyrion's escape from King's Landing. They helped each other, and it seemed that these two were only really honest with each other - but in the end, Tyrion turned on Varys, who was going to betray Daenerys. Thanks to Tyrion, Varys met death-by-dragon (even though he was right about the Queen).
Taking On Responsibility
Tyrion may have grown up as the son of a wildly powerful and wealthy house, but he was never really expected to take on responsibility. His father put up with him for the sake of family, but never intended him to become the head of the Lannister House, and so Tyrion started the series as someone who shirked responsibility. He spent his time drinking and in brothels, and sassing anyone who annoyed him - the Westerosi version of a spoiled rich kid. However, as the show progressed, he took on more and more responsibility - as Hand to the Queen, and then as Hand to Bran in the end (even though by this point, he really didn't want to have any part in ruling).
Humility And Respect
Probably the biggest change, for better or for worse, is in Tyrion's humility. In the early episodes, he is cocky, proud, and happy to buy anyone off (or drink them under the table). But his experiences over the course of the show (the loss of his position, his disfigurement, his relationship with Shae, the loss of his family, and so much more) have humbled him. He learns to respect the Starks and the Night's Watch, and to see how heavy responsibility can be - but how important it is to shoulder it anyway.